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In fluid
dynamics, Couette flow refers to the laminar flow of a viscousfluid in the space between two parallel plates,
one of which is moving relative to the other. The flow is driven by
virtue of viscous drag force acting on the fluid and the applied
pressure gradient parallel to the plates. This type of flow is
named in honor of Maurice
Marie Alfred Couette, a Professor of Physics at the French
university of Angers in the
late 19th century.

Simple conceptual
configuration

Simple Couette configuration using two infinite flat plates.

Mathematical description

Couette flow is frequently used in undergraduate physics and
engineering courses to illustrate shear-driven fluid motion[1]. The
simplest conceptual configuration finds two infinite, parallel
plates separated by a distance h. One plate, say the top
one, translates with a constant velocity u0 in
its own plane. Neglecting pressure gradients, the Navier-Stokes equations simplify to

where y is a spatial coordinate normal to the plates
and u (y) is the velocity distribution. This equation
reflects the assumption that the flow is uni-directional.
That is, only one of the three velocity components (u,v,w) is
non-trivial. If y originates at the lower plate, the
boundary conditions are u(0) = 0 and u(h) =
u0. The exact solution

can be found by integrating twice and solving for the constants
using the boundary conditions.

Constant
shear

A notable aspect of this model is that shear stress is constant throughout the
flow domain. In particular, the first derivative of the velocity,
u0/ h, is constant. (This is implied
by the straight-line profile in the figure.) According to Newton's Law of Viscosity
(Newtonian fluid), the shear stress is the product of this
expression and the (constant) fluid viscosity.

Couette flow with pressure
gradient

A more general Couette flow situation arises when a pressure
gradient is imposed in a direction parallel to the plates. The
Navier-Stokes equations, in this case, simplify to

where
is the pressure gradient parallel to the plates and μ is fluid viscosity. Integrating the above equation
twice and applying the boundary conditions (same as in the case of
Couette flow without pressure gradient) to yield the following
exact solution

The shape of the above velocity profile depends on the
dimensionless parameter

It may be noted that in the limiting case of stationary plates,
the flow is referred to as plane Poiseuille flow with a
symmetric (with reference to the horizontal mid-plane) parabolic
velocity profile.

Taylor's idealized
model

The configuration shown in the figure cannot actually be
realized, as the two plates cannot extend infinitely in the flow
direction. Sir Geoffrey Taylor was
interested in shear-driven flows created by rotating co-axial
cylinders. He reported a mathematical result in 1923 that accounts
for curvature in the flow direction having the form[2]

where C1 and C2 are
constants that depend on the rotation rates of the cylinders. (Note
that r has replaced y in this result to reflect
cylindrical rather than rectangular coordinates.) It is clear from
this equation that curvature effects no longer allow for constant
shear in the flow domain, as shown above. This model is incomplete
in that it does not account for near-wall effects in finite-width
cylinders, although it is a reasonable approximation if the width
is large compared to the space between the cylinders.
Generalizations of Taylor's basic model have also been examined.
For example, the solution for the time-dependent "start-up" process
can be expressed in terms of Bessel functions[3].

Finite-width
model

Taylor's solution accounts for the curvature inherent
in the cylindrical devices typically used to create Couette flows,
but not the finite nature of the width. A complementary
idealization accounts for finiteness, but not curvature. In the
figure above, we might think of the "boundary plate" and the
"moving plate" as the edges of two cylinders having large radii,
say R1 and R2, respectively, where
R2 is only
slightly greater than R1. In this case,
curvature can be neglected locally. The physicist/mathematician
Ratip Berker reported a mathematical solution for this
configuration in terms of a trigonometric
expansion[4]

Wendl's result for
physical devices

Actual co-axial cylinder devices used to create Couette flows
have both curvature and finite geometry. The latter gives rise to
increased drag
in the wall region. A mathematical result that accounts for both of
these aspects was given only recently by Michael Wendl[5]. His
solution takes the form of an expansion of modified (hyperbolic) Bessel
functions of the first kind.